Integrating various kinds of static and dynamic data during the reservoir modeling and simulation process has been shown to reduce the uncertainty of the simulation models, thereby improving the predictive capacity of such models, which in turn can lead to better reservoir management decisions. In this regard, the ensemble Kalman filter (EnKF) has recently generated significant attention as a promising method for conditioning reservoir simulation models to dynamic production data. Further, a recent emphasis on uncertainty quantification, closed-loop reservoir optimization and real-time monitoring has made the EnKF even more valuable, as the EnKF is particularly suited for continuous model updating, and provides an ensemble of models that can be used to approximate the posterior distribution of any output of the simulation model.
The EnKF has been recently applied and improved upon by many researchers in the petroleum industry. It was introduced to the petroleum industry by Naevdal, G., Johnsen, L. M., Aanonsen, S. I., Vefring, E. H., Reservoir Monitoring and Continuous Model Updating Using Ensemble Kalman Filter, SPE paper 84372 presented at the SPE Annual Technical Conference and Exhibition, Denver, Colo., 2003. Naevdal, G., Mannseth, T., Vefring, E. H., Near-Well Reservoir Monitoring Through Ensemble Kalman Filter, paper SPE 75235 presented at the SPE/DOE Improved Oil Recovery Symposium, Tulsa, Okla., 2002, wherein the EnKF was used to update static parameters in near-wellbore simulation models, and later also used to update permeability, pressure and saturation fields of a 2D three phase simulation model. Since then, others have modified and improved the EnKF including, Gu, Y., Oliver, D. S., History Matching of the PUNQ-S3 Reservoir Model Using the Ensemble Kalman Filter, SPE Journal, 10, 217-224, 2005, Wen, X.-H., Chen, W. H., Real-time Reservoir Model Updating Using Ensemble Kalman Filter, paper SPE 92991 presented at the SPE Reservoir Simulation Symposium, Houston, Tex., 2005, Li, G., Reynolds, A. C., An Iterative Ensemble Kalman Filter for Data Assimilation, paper SPE 109808 presented at the SPE Annual Technical Conference and Exhibition, Anaheim, Calif., 2007, Skjervheim, J. A., Evensen, G., Aanonsen, S. I., Ruud, B. O., Johansen, T. A., Incorporating 4D Seismic Data in Reservoir Simulation Models Using Ensemble Kalman Filter, SPE Journal, 12, 282-292, 2007, etc.
It is known that a key limitation of the EnKF is that it is technically appropriate only for random fields (e.g., permeability) characterized by two-point geostatistics (multi-Gaussian random fields). Application of the EnKF to complex non-Gaussian geological models such as channels systems leads to modification of these models towards Gaussianity. As a result, although a good agreement to the observed production data may be obtained from the updated models, the predictive capacity of such models may be questionable. The main reason behind this limitation is that the updated ensemble obtained using the EnKF is a linear combination of the forecasted ensemble, and the EnKF only uses the covariance and cross-covariance between the random fields (to be updated) and observations, thereby only preserving two-point statistics.
Kernel methods have recently generated significant interest in the machine learning community (Scholkopf, B., Smola, A. J., Learning with Kernels, MIT Press, Cambridge, Mass., 2002), and enable efficient nonlinear generalizations of linear algorithms. Well known examples of the application of kernel methods to linear algorithms to create nonlinear generalizations are support vector machines, kernel-based clustering, and kernel principal component analysis (Scholkopf, B., Smola, A. J., Learning with Kernels, MIT Press, Cambridge, Mass., 2002). See also Sarma, P., Durlofsky, L. J., Aziz, K., Kernel Principal Component Analysis for an Efficient, Differentiable Parameterization of Multipoint Geostatistics, Mathematical Geosciences, 40, 3-32, 2008, which describes a method that utilizes kernel PCA to parameterize non-Gaussian random fields, which could then be used with gradient-based optimization methods for efficient history matching while preserving geological realism.